Have considered a more controlled approach and skip IE altogether?
You can use an HTTP control to access the websites, post forms and download the files with complete control.

One such product is HTTP Wizard:
http://www.SeekfordSolutions.com/Products/HTTPWizard/

There are also others out there but of course I like ours the most.

You should probably at least investigate multiple ways of handling your task. You might find this way is more efficient and gives you better control. No dialogs displayed to interfere.